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The Muon Anomalous Magnetic Moment

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The Muon Anomalous Magnetic Moment UNIVERSITY OF OKLAHOMA GRADUATE COLLEGE HOMER L. DODGE DEPARTMENT OF PHYSICS AND ASTRONOMY THE MUON ANOMALOUS MAGNETIC MOMENT: A PROBE FOR THE STANDARD MODEL AND BEYOND A REPORT SUBMITTED TO THE GRADUATE FACULTY in partial fulfillment of the requirements for the SPECIALIST'S EXAMINATION By OTHMANE RIFKI Norman, Oklahoma 2014 THE MUON ANOMALOUS MAGNETIC MOMENT: A PROBE FOR THE STANDARD MODEL AND BEYOND A REPORT APPROVED FOR THE HOMER L. DODGE DEPARTMENT OF PHYSICS AND ASTRONOMY BY Dr. Brad Abbott, Chair Dr. S. Lakshmivarahan, Outside Member Dr. Mike Strauss Dr. Chung Kao Dr. Eric Abraham Table of Contents List of Tables iv List of Figures v Abstract vi 1 Introduction . .1 2 Properties of the Muon . .4 2.1 Obtaining Polarized Muons . .4 2.2 Parity Violation . .5 2.3 Relativistic Muons in a Magnetic Field . .7 3 Brookhaven g − 2 Experiment: E821 . .9 3.1 Historical Background . .9 3.2 Description of the Experimental Method . .9 3.3 Measurement of the Anomalous Angular Frequency !a ..... 11 3.4 Measurement of the Magnetic Field B .............. 14 3.5 Corrections and Systematic Errors . 15 3.6 Summary of Results from E821 . 18 4 Future Fermilab g − 2 Experiment: E989 . 19 5 The Standard Model Evaluation of the Anomaly (aµ).......... 21 5.1 Introduction . 21 5.2 The QED Contribution to aµ ................... 21 5.3 The Weak Contribution to aµ .................. 22 5.4 The Hadronic Contribution to aµ ................. 23 5.5 The SM Value of aµ ....................... 25 6 Conclusions and Prospects . 26 A The Dirac Result g = 2 27 B Spin Dynamics 30 C Muon Decay Rate 35 α D The Schwinger Term 39 2π References 43 iii List of Tables 1 Systematic errors for !a ......................... 17 2 Systematic errors for !p ......................... 18 3 Results for the anomalous precession frequency !a ........... 19 4 BNL E821 results of the anomaly aµ ................... 19 iv List of Figures 1 Pion decay . .5 2 Muon decay . .6 3 Decay electrons and asymmetry distributions in the muon rest frame . .7 4 Injection chain in the muon g − 2 experiment . 10 5 Muon spin precession in the storage ring . 11 6 Electron detection in the storage ring. 12 7 Decay electrons and asymmetry distributions in the laboratory frame. 13 8 Histogram of detected electrons . 14 9 Feynman diagram of the lowest-order contribution . 22 10 Feynman diagram of a weak contribution . 23 11 Feynman diagram of some hadronic contributions . 24 12 Different SM predictions of aµ ...................... 25 v Abstract 1 The muon is a spin- 2 charged particle characterized by an intrinsic magnetic moment with a gyromagnetic ratio, g, that is very close to 2. Its variance from 2, g − 2 referred to as the magnetic moment anomaly a = , has been determined over µ 2 the last decades to ever higher precisions in both experiment and theory. The most recent experiment (E821) was performed at Brookhaven National Laboratory and achieved a precision of 0.54 ppm, while the current theoretical evaluation stands at a precision of 0.39 ppm. However, the experimental value is higher than the predicted value by more than 3 standard deviations which suggests the possibility of new physics. A new experiment (E989) is being constructed at Fermi National Laboratory to investigate the discrepancy by reducing the experimental error to 0.14 ppm. At the same time, theory groups are working to reduce the error in aµ to match the projected experimental precision. A confirmation of the difference between experiment and theory will have an impact on new physics models in the TeV scale. The goal of this review is to describe the E821 measurement of aµ, the improvements implemented in E989, the current theoretical status in the computation of aµ, and the new physics implications. vi The closer you look the more there is to see. Friedrich Jegerlehner, The Anomalous Magnetic Moment of the Muon [1]. 1 Introduction The study of elementary particles and their interactions led to a representative mathematical formulation known as the Standard Model (SM) of particle physics. When subjected to experimental tests, the SM successfully describes three of the four fundamental forces: electromagnetic, weak, and strong interactions. On the other hand, the SM is not believed to be complete since it fails to explain a number of problems that are still facing today's physics community. First, the SM does not incorporate the fourth fundamental force of gravity. Moreover, It does not provide insight on the nature of the \invisible" matter that is holding galaxies together, which constitutes ∼ 26% of the energy density of the universe and is known as Dark Matter. In addition, the SM does not account for the different masses and mixing of the 12 leptons known as the flavor problem, and the predominance of matter over antimatter. In order to solve these problems, searches for physics not accounted for by the SM have been pursued in both experiment and theory. Any sign of significant discrepancy between experiment and theory is taken very seriously since it might lead to new insights that can reveal what is missing in our current view of the universe. Some physicists have set up experiments to look for answers to SM problems by studying high energy interactions as is pursued at the Large Hadron Collider at CERN, in the hope of observing some new particles. This led to the discovery of the Higgs boson in 2012, a central piece of the SM. Other experiments have been set up to perform detailed studies of known particles, measuring their properties to very high precisions and comparing them to theoretical calculations to both check the models and look for discrepancies. The subject of this current review is an important illustration of the latter scenario where precision tests of the magnetic moment, an intrinsic property of a spinning charged elementary particle, will be examined by comparing experiment to theory. The possible elementary charged particles that can be used to measure the 1 magnetic moment are the three spin 2 leptons: the electron e, the muon µ, and the tau τ. While these particles have the same charge and spin, they have very 1 2 2 different masses which are given by me = 0.511 MeV=c , mµ = 105.658 MeV=c , 2 and mτ = 1776.82 MeV=c . The difference in masses alters the lifetimes and decay modes of each particle. The electron is the lowest mass charged lepton and thus −6 is stable. The muon lifetime is τµ = 2:197 × 10 seconds and it decays almost 100% to an electron and two neutrinos (eνµν¯e). Taus have a much shorter lifetime −13 ττ = 2:906 × 10 seconds and a diversified decay pattern where 65% go into hadronic states (states that contain quark-antiquark pair particles such as pions) and the remainder go into leptonic states (the two possible states are muons 1The unit of mass is given in MeV/c2 according to the relation E = mc2 with the energy E given in units of MeV where 1 MeV = 1.6 × 10−13J. 1 and two neutrinos or electrons and two neutrinos) [2]. Because of its very short lifetime, the study of the tau's magnetic moment is difficult, leaving the electron and the muon as the practical candidates for measuring the magnetic moment. While the electron is the most precisely studied lepton, effects in the magnetic 2 moment sensitive to physics beyond the SM scale with powers of m` [3]. For this reason, muons are more appropriate for the study of the magnetic moment to search for new physics. The magnetic moment arises from the electric charge and the current of an elementary particle with spin. For instance, a classical calculation of a particle with mass m, and charge q, moving in a circular orbit of radius r, with velocity −!v , shows that its magnetic moment −!µ is related to its −! orbital angular momentum ( L = m−!r × −!v ) by the relation: q −! −!µ = L: (1) 2mc In quantum mechanics, the magnetic moment is an intrinsic property of a particle with spin. Both the magnetic moment and the orbital angular momentum are promoted to operators in order to give the correct quantum mechanical representation. While Equation (1) is still valid in describing the orbital angular −! momentum L , the spin magnetic moment requires a modification by a factor g that is very close to 2. The corrected equation is given by q −! −!µ = g S; (2) 2mc where g is called the gyromagnetic ratio, the Lande g-factor, or the g-factor, and q is the charge given in units of the fundamental charge e, where q = −e for a lepton particle (negative muon) and q = +e for a lepton antiparticle (positive −! muon). S is the spin operator −! S = ~−!σ ; (3) 2 where σi are the Pauli spin matrices. The result g = 2 was first obtained by Dirac in 1928 when he generalized the Schr¨odingerequation to incorporate special relativity (See Appendix A). With the development of the quantum mechanical description of electromagnetism known as quantum electrodynamics (QED), g was found to differ from 2 by an anomaly a`, known as the magnetic moment anomaly, or the anomaly for short, such that: g` = 2(1 + a`). The anomaly is then g − 2 a = ` : (4) ` 2 In this equation, the g − 2 factor appears! The factor g − 2 is incorporated in the title of all experiments that measure the magnetic moment of the muon and it is the focus of this review. In addition to the quantum fluctuations of the electromagnetic field described by QED, quantum fluctuations due to heavier particles such as the weak gauge 2 bosons (W± and Z bosons) and hadrons (for example quark-antiquark pairs such as pions) also contribute to the anomaly.
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